Some Continuum Percolation Results

  • Salvatore Torquato
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 16)


The intent of the present chapter is to derive and discuss some basic results and specific developments in continuum percolation theory. We will begin with a discussion of exact results for cluster statistics and other percolation descriptors for a prototypical model of continuum percolation, namely, identical overlapping spheres in d dimensions. Subsequently, we will describe an Ornstein-Zernike formalism to find the pair-connectedness function P 2(r) for general isotropic models of continuum percolation. The reader should note the beautiful correspondence of this theory to the Ornstein-Zernike formalism for the total correlation function h(r) of equilibrium (or thermal) systems discussed in Chapter 3. This will be followed by a discussion of various approximation schemes to close the resulting integral equation, including the Percus-Yevick approximation. The next topic will be the two-point cluster function C 2(r). First we will present an exact series representation of C 2(r) for dispersions and then discuss its analytical evaluation for certain models. The chapter will conclude with a presentation of percolation thresholds for overlapping sphere systems, overlapping particles of nonspherical shape, and interacting particle systems. The reader is referred to Meester and Roy (1996) for a more mathematical treatment of continuum percolation.


Percolation Threshold Interact Particle System Average Coordination Number Continuum Percolation Total Correlation Function 
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Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Salvatore Torquato
    • 1
  1. 1.Department of Chemistry and Princeton Materials InstitutePrinceton UniversityPrincetonUSA

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